Paper by Erik D. Demaine

Reference:

Erik D. Demaine and MohammadTaghi Hajiaghayi, “Graphs Excluding a Fixed Minor have Grids as Large as Treewidth, with Combinatorial and Algorithmic Applications through Bidimensionality”, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), Vancouver, British Columbia, Canada, January 23–25, 2005, pages 682–689.

Abstract:

We prove that any H-minor-free graph, for a fixed graph H, of
treewidth w has an
Ω(w) × Ω(w) grid graph as a minor.
Thus grid minors suffice to certify that H-minor-free graphs have large
treewidth, up to constant factors. This strong relationship was previously
known for the special cases of planar graphs and bounded-genus graphs, and is
known not to hold for general graphs. The approach of this paper can be
viewed more generally as a framework for extending combinatorial results on
planar graphs to hold on H-minor-free graphs for any
fixed H. Our result has many combinatorial consequences on
bidimensionality theory, parameter-treewidth bounds, separator theorems, and
bounded local treewidth; each of these combinatorial results has several
algorithmic consequences including subexponential fixed-parameter algorithms
and approximation algorithms.